Detection of bifurcation points

The bifurcations are detected during a call to br = continuation(prob, alg, contParams::ContinuationPar;kwargs...) by turning on the following flags:

• contParams.detect_bifurcation = 2 (for eigenvalues based bifurcations)
• contParams.detect_event = 2 (for other bifurcations like inclinations)

Precise detection of bifurcation points using Bisection

Note that the bifurcation points detected when detect_bifurcation = 2 can be rather crude localization of the true bifurcation points. Indeed, we only signal that, in between two continuation steps which can be large, a (several) bifurcation has been detected. Hence, we only have a rough idea of where the bifurcation is located, unless your dsmax is very small... This can be improved as follows.

If you choose detect_bifurcation = 3, a bisection algorithm is used to locate the bifurcation points more precisely. It means that we recursively track down the change in stability. Some options in ContinuationPar control this behavior:

• n_inversion: number of sign inversions in the bisection algorithm
• max_bisection_steps maximum number of bisection steps
• tol_bisection_eigenvalue tolerance on real part of eigenvalue to detect bifurcation points in the bisection steps

List of detected bifurcation points

You can detect the following codim 2 bifurcation points by using the option detect_codim2_bifurcation in the method continuation. Under the hood, the detection of these bifurcations is done by using Event detection as explained in Event Handling.

We refer to ^[DeWitte] for a description of the bifurcations.

Inclination-Flip with respect to the Stable / Unstable manifold is not yet detected.

References